Non-convex sparse optimization-based impact force identification with limited vibration measurements

Abstract

Impact force identification is important for structure health monitoring especially in applications involving composite structures. Different from the traditional direct measurement method, the impact force identification technique is more cost effective and feasible because it only requires a few sensors to capture the system response and infer the information about the applied forces. This technique enables the acquisition of impact locations and time histories of forces, aiding in the rapid assessment of potentially damaged areas and the extent of the damage. As a typical inverse problem, impact force reconstruction and localization is a challenging task, which has led to the development of numerous methods aimed at obtaining stable solutions. The classical {\ell }_2 regularization method often struggles to generate sparse solutions. When solving the under-determined problem, {\ell }_2 regularization often identifies false forces in non-loaded regions, interfering with the accurate identification of the true impact locations. The popular {\ell }_1 sparse regularization, while promoting sparsity, underestimates the amplitude of impact forces, resulting in biased estimations. To alleviate such limitations, a novel non-convex sparse regularization method that uses the non-convex {\ell }_{1-2} penalty, which is the difference of the {\ell }_1 and {\ell }_2 norms, as a regularizer, is proposed in this paper. The principle of alternating direction method of multipliers (ADMM) is introduced to tackle the non-convex model by facilitating the decomposition of the complex original problem into easily solvable subproblems. The proposed method named {\ell }_{1-2}\text{-ADMM} is applied to solve the impact force identification problem with unknown force locations, which can realize simultaneous impact localization and time history reconstruction with an under-determined, sparse sensor configuration. Simulations and experiments are performed on a composite plate to verify the identification accuracy and robustness with respect to the noise of the {\ell }_{1-2}\text{-ADMM} method. Results indicate that compared with other existing regularization methods, the {\ell }_{1-2}\text{-ADMM} method can simultaneously reconstruct and localize impact forces more accurately, facilitating sparser solutions, and yielding more accurate results.

Key words： impact force identification; inverse problem; sparse regularization; under-determined condition; alternating direction method of multipliers

Nomenclature

Abbreviations
ADMM	Alternating direction method of multipliers
BVID	Barely visible impact damage
GRE	Global relative error
IRF	Impulse response function
LE	Localization error
LRE	Local relative error
MC-GSURE	Monte Carlo generalized stein unbiased risk estimate
PRE	Peak relative error
SISO	Single-input single-output
SNR	Signal-noise ratio
Variables
a(t)	Impulse response function
aij(t)	Impulse response function between the output position i and the input location j
aij(ω)	Frequency response function between the output position i and the input location j
A	Transfer matrix of the multiple-input multiple-output dynamic system
As	Transfer matrix of the single-input single-output dynamic system
B	Amplitude of the Gaussian-shaped impact force
c	All-ones vector
C	Damping matrix
e	Random noise in measurements
E	Elastic modulus
fi	Elements in the vector f
f(t)	Impact force excitation function
f	Force vector of the multiple-input multiple-output dynamic system
\hat{\mathit{f}}	Estimated vector of f
fp	Actual force vector at the impact position p
{\hat{\mathit{f}}}_p	Estimated force vector at the impact position p
{\hat{\mathit{f}}}_{\text{ML}}	Maximum likelihood estimation of the vector f
fs	Force vector of the single-input single-output dynamic system
g(f)	General representation function of penalty terms
G	Shear modulus
h	An additional vector for variable splitting
\hat{\mathit{h}}	Estimated vector of h
i	Looping variable within the summation operation
I	Identity matrix
k	Number of iterations
K	Stiffness matrix
m	Number of measurement responses
M	Mass matrix
n	Number of impact force excitations
nf	Total number of potential impact force locations
N	Data length of the discretized impulse response function
Nmax	Maximum number of iterations
O(nN)	Computational complexity of n × N
p	Serial number of the location subjected to impact force
q	Norm parameter defined in {\mathbb{R}}^{+\ast }
Q	Projection matrix
r	Gaussian white noise vector
\mathcal{R}(\mathit{f})	General expression for calculating the norm of vector f
s(t)	System response
s	Response vector of the multiple-input multiple-output dynamic system
\stackrel{\mathbf{~}}{\mathit{s}}	Noisy response vector
si	Response vector at a certain position i
ss	Response vector of the single-input single-output dynamic system
t	Time
t0	Occurrence time instant of the impact
Δt	Sampling interval
T	Impact duration
u	Sufficient statistic of the model Eq. (8)
wmax	Element with the largest absolute value in the vector·w
w	Intermediate vector defined as w = f (k+1) + z(k)
xλ(u)	Solution result of Eq. (8) when f = u
y(f)	Proximal operator
δ	Small positive parameter
δ	Lagrange multiplier vector
ε	Iteration termination threshold
λ	Regularization parameter
ρ	A positive penalty parameter
σ	Standard deviation of the vector \mathit{s}
σn	Standard deviation of noise in the measurements
τ	Time delayed operator
ν	Possion’s ratio
Γ	Threshold value defined as Γ = λ/ρ

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52075414 and 52241502), and China Postdoctoral Science Foundation (Grant No. 2021M702595).

Conflict of Interest

The authors declare that they have no conflict of interest.

References

1 Wu J , Xu X B , Liu C , Deng C , Shao X Y . Lamb wave-based damage detection of composite structures using deep convolutional neural network and continuous wavelet transform. Composite Structures, 2021, 276: 114590

2 Gu X J , Su X Z , Wang J , Xu Y J , Zhu J H , Zhang W H . Improvement of impact resistance of plain-woven composite by embedding superelastic shape memory alloy wires. Frontiers of Mechanical Engineering, 2020, 15(4): 547–557

3 Han D , Jia X , Zhang H J , Gao X G , Han X , Sun L , Zheng Z K , Zhang L , Wang F , Song Y D . Foreign object damage and post-impact tensile behavior of plain-woven SiC/SiC composites. Composite Structures, 2022, 295: 115767

4 Singh T , Sehgal S . Structural health monitoring of composite materials. Archives of Computational Methods in Engineering, 2022, 29(4): 1997–2017

5 Chen X F , Wang S B , Qiao B J , Chen Q . Basic research on machinery fault diagnostics: past, present, and future trends. Frontiers of Mechanical Engineering, 2018, 13(2): 264–291

6 Liu R X , Dobriban E , Hou Z C , Qian K . Dynamic load identification for mechanical systems: a review. Archives of Computational Methods in Engineering, 2022, 29(2): 831–863

7 WangLLiuY RXuH Y. Review: recent developments in dynamic load identification for aerospace vehicles considering multi-source uncertainties. Transactions of Nanjing University of Aeronautics & Astronautics, 2021, 38(2): 271–287 (in Chinese)

8 Jacquelin E , Bennani A , Hamelin P . Force reconstruction: analysis and regularization of a deconvolution problem. Journal of Sound and Vibration, 2003, 265(1): 81–107

9 Li Q F , Lu Q H . Impact localization and identification under a constrained optimization scheme. Journal of Sound and Vibration, 2016, 366: 133–148

10 Yan G , Sun H , Büyüköztürk O . Impact load identification for composite structures using Bayesian regularization and unscented Kalman filter. Structural Control and Health Monitoring, 2017, 24(5): e1910

11 Aucejo M , De Smet O , Deü J F . On a space-time regularization for force reconstruction problems. Mechanical Systems and Signal Processing, 2019, 118: 549–567

12 RezayatANassiriVDe PauwBErtveldtJVanlanduitSGuillaumeP. Identification of dynamic forces using group-sparsity in frequency domain. Mechanical Systems and Signal Processing, 2016, 70–71: 756–768

13 Candelieri A , Archetti F . Sparsifying to optimize over multiple information sources: an augmented gaussian process based algorithm. Structural and Multidisciplinary Optimization, 2021, 64(1): 239–255

14 Figueiredo M A T , Nowak R D , Wright S J . Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 586–597

15 Kim S J , Koh K , Lustig M , Boyd S , Gorinevsky D . An interior-point method for large-scale ℓ1-regularized least squares. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 606–617

16 Daubechies I , Defrise M , De Mol C . An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 2004, 57(11): 1413–1457

17 GinsbergDFritzenC P. New approach for impact detection by finding sparse solution. In: Proceedings of ISMA. Leuven: Katholieke Universiteit Leuven, 2014, 2043–2056

18 Qiao B J , Zhang X W , Gao J W , Chen X F . Impact-force sparse reconstruction from highly incomplete and inaccurate measurements. Journal of Sound and Vibration, 2016, 376: 72–94

19 Aucejo M , De Smet O . A space-frequency multiplicative regularization for force reconstruction problems. Mechanical Systems and Signal Processing, 2018, 104: 1–18

20 Pan C D , Chen Z P . Elimination of accelerometer mass loading effects in sparse identification of impact forces. Mechanical Systems and Signal Processing, 2023, 191: 110178

21 Xu Z B , Chang X Y , Xu F M , Zhang H . L_1/2 regularization: a thresholding representation theory and a fast solver. IEEE Transactions on Neural Networks and Learning Systems, 2012, 23(7): 1013–1027

22 TrehanD. Non-convex optimization: a review. In: Proceedings of 2020 the 4th International Conference on Intelligent Computing and Control Systems. Madurai: IEEE, 2020, 418–423

23 Zha Z Y , Wen B H , Yuan X , Ravishankar S , Zhou J T , Zhu C . Learning nonlocal sparse and low-rank models for image compressive sensing: nonlocal sparse and low-rank modeling. IEEE Signal Processing Magazine, 2023, 40(1): 32–44

24 Zha Z Y , Wen B H , Yuan X , Zhou J T , Zhu C . Image restoration via reconciliation of group sparsity and low-rank models. IEEE Transactions on Image Processing, 2021, 30: 5223–5238

25 ChartrandRYinW T. Nonconvex sparse regularization and splitting algorithms. In: Glowinski R, Osher S J, Yin W T, eds. Splitting Methods in Communication, Imaging, Science, and Engineering. Cham: Springer, 2016, 237–249

26 Qiao B J , Ao C Y , Mao Z , Chen X F . Non-convex sparse regularization for impact force identification. Journal of Sound and Vibration, 2020, 477: 115311

27 Aucejo M , De Smet O . A generalized multiplicative regularization for input estimation. Mechanical Systems and Signal Processing, 2021, 157: 107637

28 Liu J J , Qiao B J , Wang Y N , He W F , Chen X F . Non-convex sparse regularization via convex optimization for impact force identification. Mechanical Systems and Signal Processing, 2023, 191: 110191

29 Esser E , Lou Y F , Xin J . A method for finding structured sparse solutions to nonnegative least squares problems with applications. SIAM Journal on Imaging Sciences, 2013, 6(4): 2010–2046

30 Yin P H , Lou Y F , He Q , Xin J . Minimization of ℓ₁₋₂ for compressed sensing. SIAM Journal on Scientific Computing, 2015, 37(1): A536–A563

31 Sun H , Feng D M , Liu Y , Feng M Q . Statistical regularization for identification of structural parameters and external loadings using state space models. Computer-Aided Civil and Infrastructure Engineering, 2015, 30(11): 843–858

32 Ginsberg D , Ruby M , Fritzen C P . Load identification approach based on basis pursuit denoising algorithm. Journal of Physics: Conference Series, 2015, 628(1): 012030

33 Boyd S , Parikh N , Chu E , Peleato B , Eckstein J . Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 2011, 3(1): 1–122

34 MeirovitchL. Elements of Vibration Analysis. New York: McGraw-Hill Book Co., 1986

35 Jankowski L . Off-line identification of dynamic loads. Structural and Multidisciplinary Optimization, 2009, 37(6): 609–623

36 ZhaZ YLiuXHuangX HShiH LXuY YWangQTangLZhangX G. Analyzing the group sparsity based on the rank minimization methods. In: Proceedings of 2017 IEEE International Conference on Multimedia and Expo. Hong Kong: IEEE, 2017, 883–888

37 Zha Z Y , Zhang X G , Wu Y , Wang Q , Liu X , Tang L , Yuan X . Non-convex weighted ℓp nuclear norm based ADMM framework for image restoration. Neurocomputing, 2018, 311: 209–224

38 Li X W , Zhao H T , Chen Z , Wang Q B , Chen J A , Duan D P . Force identification based on a comprehensive approach combining Taylor formula and acceleration transmissibility. Inverse Problems in Science and Engineering, 2018, 26(11): 1612–1632

39 Wang L , Liu J K , Lu Z R . Bandlimited force identification based on sinc-dictionaries and Tikhonov regularization. Journal of Sound and Vibration, 2020, 464: 114988

40 Qiao B J , Liu J J , Liu J X , Yang Z B , Chen X F . An enhanced sparse regularization method for impact force identification. Mechanical Systems and Signal Processing, 2019, 126: 341–367

41 Lou Y F , Yan M . Fast L₁–L₂ minimization via a proximal operator. Journal of Scientific Computing, 2018, 74(2): 767–785

42 Chang X T , Yan Y J , Wu Y F . Study on solving the ill-posed problem of force load reconstruction. Journal of Sound and Vibration, 2019, 440: 186–201

43 Cao W F , Sun J , Xu Z B . Fast image deconvolution using closed-form thresholding formulas of L_q (q = 1/2, 2/3) regularization. Journal of Visual Communication and Image Representation, 2013, 24(1): 31–41

44 Zha Z Y , Yuan X , Wen B H , Zhou J T , Zhang J C , Zhu C . A benchmark for sparse coding: when group sparsity meets rank minimization. IEEE Transactions on Image Processing, 2020, 29: 5094–5109

45 Eldar Y C . Generalized SURE for exponential families: applications to regularization. IEEE Transactions on Signal Processing, 2009, 57(2): 471–481

46 Donoho D L . De-noising by soft-thresholding. IEEE Transactions on Information Theory, 1995, 41(3): 613–627

47 Ramani S , Blu T , Unser M . Monte-carlo sure: a black-box optimization of regularization parameters for general denoising algorithms. IEEE Transactions on Image Processing, 2008, 17(9): 1540–1554